Define the set of unitic univariate polynomials, as the polynomials with
an invertible leading coefficient. This is not a standard concept but
is useful to us as the set of polynomials which can be used as the
divisor in the polynomial division theorem ply1divalg19518. (Contributed
by Stefan O'Rear, 28-Mar-2015.)

Define the quotient of two univariate polynomials, which is guaranteed
to exist and be unique by ply1divalg19518. We actually use the reversed
version for better harmony with our divisibility df-dvdsr15418.
(Contributed by Stefan O'Rear, 28-Mar-2015.)

Uniqueness of a quotient in a polynomial division. For polynomials
such that and the
leading coefficient of
is not a zero divisor, there is at most one polynomial which
satisfies
where the degree of is
less
than the degree of . (Contributed by Stefan O'Rear,
26-Mar-2015.) (Revised by NM, 17-Jun-2017.)

The division algorithm for univariate polynomials over a ring. For
polynomials such that and the leading
coefficient of is a unit, there are unique polynomials and
such
that the degree of is
less than the
degree of .
(Contributed by Stefan O'Rear, 27-Mar-2015.)

The polynomial remainder theorem, or little Bézout's theorem
(by contrast to the regular Bézout's theorem bezout12716). If a
polynomial
is divided by the linear factor , the
remainder is equal to , the evaluation of the polynomial
at
(interpreted as a constant polynomial). (Contributed by
Mario Carneiro, 12-Jun-2015.)

The one-sided fundamental theorem of algebra. A polynomial of degree
has at most
roots. Unlike the
real fundamental theorem
fta20312, which is only true in and other algebraically closed
fields, this is true in any integral domain. (Contributed by Mario
Carneiro, 12-Jun-2015.)

The set of polynomials is unaffected by the addition of zero. (This is
built into the definition because all higher powers of a polynomial are
effectively zero, so we require that the coefficient field contain zero
to simplify some of our closure theorems.) (Contributed by Mario
Carneiro, 17-Jul-2014.)

Lemma for plyeq019588. If is the coefficient function for a
nonzero polynomial such that
for every
and is the nonzero leading coefficient, then the function
is
a sum of powers of , and so
the limit of this function as is the constant term,
. But everywhere, so this limit is
also
equal to zero so that , a
contradiction. (Contributed
by Mario Carneiro, 22-Jul-2014.)

If a polynomial is zero at every point (or even just zero at the
positive integers), then all the coefficients must be zero. This is the
basis for the method of equating coefficients of equal polynomials, and
ensures that df-coe19567 is well-defined. (Contributed by Mario
Carneiro,
22-Jul-2014.)